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Méthodes d'inférence exactes pour un modèle de régression avec erreurs AR(2) gaussiennes


  • Jean-Marie Dufour
  • Malika Neifar


In this paper, we consider a linear regression model with Gaussian autoregressive errors of order p = 2, which may be nonstationary. Exact inference methods (tests and confidence region) are developed for the autoregressive parameters and the regression coefficients. We generalize the method proposed in Dufour (1990) for linear regression models with autoregressive errors of order p = 1: The proposed approach consists in three stages. First, we build an exact confidence set for the complete vector of the autoregressive coefficients (varphi). This region is obtained by inverting independence tests for model errors after the model has been transformed to get independent errors under the null hypothesis. The independence tests are based on combining tests for the presence of autocorrelation at lags one and two. Exploiting the duality between tests and confidence sets, an exact confidence set is then built by finding the set of autoregressive parameter values which are not rejected (test inversion). Second, using this confidence set for (varphi), simultaneous confidence sets for the autoregressive parameters and regression coefficients are obtained. Finally, marginal confidence intervals for the regression coefficients are derived using a projection approach. We also propose generalized bounds tests for the regression parameters. These methods are applied to time series models of the U.S. money stock (M2) and GNP deflator. Ce texte propose des méthodes d'inférence exactes (tests et régions de confiance) sur des modèles de régression linéaires avec erreurs autocorrélées suivant un processus autorégressif d'ordre deux [AR(2)], qui peut être non-stationnaire. L'approche proposée est une généralisation de celle décrite dans Dufour (1990) pour un modèle de régression avec erreurs AR(1) et comporte trois étapes. Premièrement, on construit une région de confiance exacte pour le vecteur des coefficients du processus autorégressif (varphi). Cette région est obtenue par inversion de tests d'indépendance des erreurs sur une forme transformée du modèle contre des alternatives de dépendance aux délais un et deux. Deuxièmement, en exploitant la dualité entre tests et régions de confiance (inversion de tests), on détermine une région de confiance conjointe pour le vecteur (varphi) et un vecteur d'intérêt (gamma) de combinaisons linéaires des coefficients de régression du modèle. Troisièmement, par une méthode de projection, on obtient des intervalles de confiance «marginaux» ainsi que des tests à bornes exacts pour les composantes de (varphi). Ces méthodes sont appliquées à des modèles du stock de monnaie (M2) et du niveau des prix (indice implicite du PNB) américains.

Suggested Citation

  • Jean-Marie Dufour & Malika Neifar, 2003. "Méthodes d'inférence exactes pour un modèle de régression avec erreurs AR(2) gaussiennes," CIRANO Working Papers 2003s-54, CIRANO.
  • Handle: RePEc:cir:cirwor:2003s-54

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    References listed on IDEAS

    1. Wallis, Kenneth F, 1972. "Testing for Fourth Order Autocorrelation in Qtrly Regression Equations," Econometrica, Econometric Society, vol. 40(4), pages 617-636, July.
    2. Dufour, Jean-Marie & King, Maxwell L., 1991. "Optimal invariant tests for the autocorrelation coefficient in linear regressions with stationary or nonstationary AR(1) errors," Journal of Econometrics, Elsevier, vol. 47(1), pages 115-143, January.
    3. Savin, N.E., 1984. "Multiple hypothesis testing," Handbook of Econometrics,in: Z. Griliches† & M. D. Intriligator (ed.), Handbook of Econometrics, edition 1, volume 2, chapter 14, pages 827-879 Elsevier.
    4. Nankervis, J. C. & Savin, N. E., 1985. "Testing the autoregressive parameter with the t statistic," Journal of Econometrics, Elsevier, vol. 27(2), pages 143-161, February.
    5. Ansley, Craig F. & Kohn, Robert & Shively, Thomas S., 1992. "Computing p-values for the generalized Durbin-Watson and other invariant test statistics," Journal of Econometrics, Elsevier, vol. 54(1-3), pages 277-300.
    6. King, Maxwell L., 1985. "A point optimal test for autoregressive disturbances," Journal of Econometrics, Elsevier, vol. 27(1), pages 21-37, January.
    7. Jean-Marie Dufour & Jan F. Kiviet, 1998. "Exact Inference Methods for First-Order Autoregressive Distributed Lag Models," Econometrica, Econometric Society, vol. 66(1), pages 79-104, January.
    8. Nelson, Charles R. & Plosser, Charles I., 1982. "Trends and random walks in macroeconmic time series : Some evidence and implications," Journal of Monetary Economics, Elsevier, vol. 10(2), pages 139-162.
    9. King, M. L., 1981. "The alternative Durbin-Watson test : An assessment of Durbin and Watson's choice of test statistic," Journal of Econometrics, Elsevier, vol. 17(1), pages 51-66, September.
    10. DeJong, David N. & Nankervis, John C. & Savin, N. E. & Whiteman, Charles H., 1992. "The power problems of unit root test in time series with autoregressive errors," Journal of Econometrics, Elsevier, vol. 53(1-3), pages 323-343.
    11. Andrews, Donald W K, 1993. "Tests for Parameter Instability and Structural Change with Unknown Change Point," Econometrica, Econometric Society, vol. 61(4), pages 821-856, July.
    12. Park, Rolla Edward & Mitchell, Bridger M., 1980. "Estimating the autocorrelated error model with trended data," Journal of Econometrics, Elsevier, vol. 13(2), pages 185-201, June.
    13. Dufour, Jean-Marie, 1990. "Exact Tests and Confidence Sets in Linear Regressions with Autocorrelated Errors," Econometrica, Econometric Society, vol. 58(2), pages 475-494, March.
    14. Miyazaki, Shigetaka & Griffiths, William E., 1984. "The properties of some covariance matrix estimators in linear models with AR(1) errors," Economics Letters, Elsevier, vol. 14(4), pages 351-356.
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    More about this item


    AR(2) errors; exact test; confidence set; induced test; generalized bounds test; projection method; test inversion; money stock; M2; price level; régression linéaire; autocorrélation; AR(2); test exact; région de confiance exacte; test induit; test à borne généralisé; projection; masse monétaire; M2; niveau des prix;

    JEL classification:

    • M2 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Business Economics
    • M2 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Business Economics


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